direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C12×F5, C60⋊4C4, C20⋊2C12, C15⋊3C42, Dic5⋊2C12, C5⋊(C4×C12), D5.(C2×C12), C2.2(C6×F5), (C4×D5).6C6, (C2×F5).2C6, (C6×F5).4C2, C6.17(C2×F5), C10.3(C2×C12), C30.17(C2×C4), (C3×Dic5)⋊6C4, D10.4(C2×C6), (D5×C12).13C2, (C6×D5).23C22, (C3×D5).3(C2×C4), SmallGroup(240,113)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — C12×F5 |
Generators and relations for C12×F5
G = < a,b,c | a12=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >
Subgroups: 156 in 60 conjugacy classes, 36 normal (20 characteristic)
C1, C2, C2, C3, C4, C4, C22, C5, C6, C6, C2×C4, D5, C10, C12, C12, C2×C6, C15, C42, Dic5, C20, F5, D10, C2×C12, C3×D5, C30, C4×D5, C2×F5, C4×C12, C3×Dic5, C60, C3×F5, C6×D5, C4×F5, D5×C12, C6×F5, C12×F5
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, C12, C2×C6, C42, F5, C2×C12, C2×F5, C4×C12, C3×F5, C4×F5, C6×F5, C12×F5
►On 60 points
(1 2 3 4 5 6 7 8 9 10 11 12)(13 14 15 16 17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32 33 34 35 36)(37 38 39 40 41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56 57 58 59 60)
(1 17 54 33 43)(2 18 55 34 44)(3 19 56 35 45)(4 20 57 36 46)(5 21 58 25 47)(6 22 59 26 48)(7 23 60 27 37)(8 24 49 28 38)(9 13 50 29 39)(10 14 51 30 40)(11 15 52 31 41)(12 16 53 32 42)
(1 7)(2 8)(3 9)(4 10)(5 11)(6 12)(13 56 39 35)(14 57 40 36)(15 58 41 25)(16 59 42 26)(17 60 43 27)(18 49 44 28)(19 50 45 29)(20 51 46 30)(21 52 47 31)(22 53 48 32)(23 54 37 33)(24 55 38 34)
G:=sub<Sym(60)| (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,17,54,33,43)(2,18,55,34,44)(3,19,56,35,45)(4,20,57,36,46)(5,21,58,25,47)(6,22,59,26,48)(7,23,60,27,37)(8,24,49,28,38)(9,13,50,29,39)(10,14,51,30,40)(11,15,52,31,41)(12,16,53,32,42), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,56,39,35)(14,57,40,36)(15,58,41,25)(16,59,42,26)(17,60,43,27)(18,49,44,28)(19,50,45,29)(20,51,46,30)(21,52,47,31)(22,53,48,32)(23,54,37,33)(24,55,38,34)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12)(13,14,15,16,17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32,33,34,35,36)(37,38,39,40,41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56,57,58,59,60), (1,17,54,33,43)(2,18,55,34,44)(3,19,56,35,45)(4,20,57,36,46)(5,21,58,25,47)(6,22,59,26,48)(7,23,60,27,37)(8,24,49,28,38)(9,13,50,29,39)(10,14,51,30,40)(11,15,52,31,41)(12,16,53,32,42), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12)(13,56,39,35)(14,57,40,36)(15,58,41,25)(16,59,42,26)(17,60,43,27)(18,49,44,28)(19,50,45,29)(20,51,46,30)(21,52,47,31)(22,53,48,32)(23,54,37,33)(24,55,38,34) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12),(13,14,15,16,17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32,33,34,35,36),(37,38,39,40,41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56,57,58,59,60)], [(1,17,54,33,43),(2,18,55,34,44),(3,19,56,35,45),(4,20,57,36,46),(5,21,58,25,47),(6,22,59,26,48),(7,23,60,27,37),(8,24,49,28,38),(9,13,50,29,39),(10,14,51,30,40),(11,15,52,31,41),(12,16,53,32,42)], [(1,7),(2,8),(3,9),(4,10),(5,11),(6,12),(13,56,39,35),(14,57,40,36),(15,58,41,25),(16,59,42,26),(17,60,43,27),(18,49,44,28),(19,50,45,29),(20,51,46,30),(21,52,47,31),(22,53,48,32),(23,54,37,33),(24,55,38,34)]])
C12×F5 is a maximal subgroup of
C30.3C42 D12⋊2F5 D60⋊5C4 (C4×S3)⋊F5
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | 4B | 4C | ··· | 4L | 5 | 6A | 6B | 6C | 6D | 6E | 6F | 10 | 12A | 12B | 12C | 12D | 12E | ··· | 12X | 15A | 15B | 20A | 20B | 30A | 30B | 60A | 60B | 60C | 60D |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | ··· | 4 | 5 | 6 | 6 | 6 | 6 | 6 | 6 | 10 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 15 | 15 | 20 | 20 | 30 | 30 | 60 | 60 | 60 | 60 |
| size | 1 | 1 | 5 | 5 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 1 | 1 | 5 | 5 | 5 | 5 | 4 | 1 | 1 | 1 | 1 | 5 | ··· | 5 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C12 | C12 | C12 | F5 | C2×F5 | C3×F5 | C4×F5 | C6×F5 | C12×F5 |
| kernel | C12×F5 | D5×C12 | C6×F5 | C4×F5 | C3×Dic5 | C60 | C3×F5 | C4×D5 | C2×F5 | Dic5 | C20 | F5 | C12 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 2 | 4 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 2 | 4 |
Matrix representation of C12×F5 ►in GL5(𝔽61)
| 50 | 0 | 0 | 0 | 0 |
| 0 | 13 | 0 | 0 | 0 |
| 0 | 0 | 13 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 60 |
| 0 | 1 | 0 | 0 | 60 |
| 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 1 | 60 |
| 11 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(61))| [50,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13,0,0,0,0,0,13],[1,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,60,60,60,60],[11,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0] >;
C12×F5 in GAP, Magma, Sage, TeX
C_{12}\times F_5 % in TeX
G:=Group("C12xF5"); // GroupNames label
G:=SmallGroup(240,113);
// by ID
G=gap.SmallGroup(240,113);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-5,72,151,3461,599]);
// Polycyclic
G:=Group<a,b,c|a^12=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations